using cosh(x) ? - Printable Version +- Tetration Forum ( https://math.eretrandre.org/tetrationforum)+-- Forum: Tetration and Related Topics ( https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1)+--- Forum: Mathematical and General Discussion ( https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3)+--- Thread: using cosh(x) ? ( /showthread.php?tid=718) |

using cosh(x) ? - tommy1729 - 11/19/2011
no , im not kidding , apart from my sinh method im considering a cosh method. although it might be more complicated or speculative. it might be insightfull to plot things. consider exp(1/e)<b<sqrt(e) f(x) - f^[-1](x) = 2*cosh(ln(b)*x) for variable x and fixed b ( base ). i bet you can find more functional equations from this one. for instance to express f^[-1](x) in terms of f(x) and elementary functions. the solution is unique for R-> R. replace x by f(x) and let g(x) be the super of f(x). then we get a difference equation for g(x). we solve that difference equation , now we go from g(x) to f(x) or we find the taylor for f(x) from the equation ( or both , in programming ). similar to the sinh method : since f is close to 2*cosh(ln(b)*x)) and 2*cosh(ln(b)*x) is close to b^x , we have a new method. RE: using cosh(x) ? - tommy1729 - 11/20/2011
also , we finally have boundaries. for large x , f^[1/2](x) + f^[1/2](ln_b(x)) is a good bound for tet(b,1/2,x). RE: using cosh(x) ? - tommy1729 - 11/20/2011
hmm i was thinking about f(x) - f[-1](x) = 2*sinh(ln(b)*x). for approximating (b^x)^[theta] for bases b with eta < b < sqrt(e) and x > e^e. RE: using cosh(x) ? - tommy1729 - 11/20/2011
a possible improvement would be f(x) - f^[-1](x) = 2*sinh(ln(b)*x) + (e-2)*x/e. f(0) = 0 tommy1729 RE: using cosh(x) ? - sheldonison - 11/20/2011
(11/19/2011, 11:56 PM)tommy1729 Wrote: no , im not kidding , apart from my sinh method im considering a cosh method.What is the fixed point for this method? The fixed point for 2sinh was zero. RE: using cosh(x) ? - tommy1729 - 11/20/2011
(11/20/2011, 03:18 PM)sheldonison Wrote:(11/19/2011, 11:56 PM)tommy1729 Wrote: no , im not kidding , apart from my sinh method im considering a cosh method.What is the fixed point for this method? The fixed point for 2sinh was zero. the fixed point is pi/2 i for the cosh method. for the ' new ' sinh method it is also 0. |